Higher-order Boltzmann equation approximations 70

In the previous sections, we found the first-order perturbation f⁽¹⁾to the distribution function, and its corresponding first-order moment perturba-tionsσ⁽¹⁾andq⁽¹⁾.

Of course, it is possible to go further, finding f⁽²⁾, σ⁽²⁾, and q⁽²⁾. While we get the Euler model by assuming f f⁽⁰⁾ and the Navier-Stokes-Fourier model by assuming f f⁽⁰⁾+ f⁽¹⁾, the assumption of

f f⁽⁰⁾+f⁽¹⁾+f⁽²⁾gives us an even more detailed picture, called the Burnett model. A far tougher derivation leads to moment perturbations

Burnett model The fluid model found by taking Chapman-Enskog one step further than the Navier-Stokes-Fourier model

σ⁽²⁾andq⁽²⁾ which contain several new terms and several new transport coefficients that cannot be predicted by continuum theory [57, Ch. 15].

Since f⁽²⁾/ f⁽⁰⁾ is O(Kn²), the extra terms of the Burnett model are usually negligible, as Kn 1 in most flows of interest. However, the differences between the Navier-Stokes-Fourier model and the Burnett model are significant for the propagation of sound waves at very high

fre-3.8 The Chapman-Enskog expansion 71

quencies, where the acoustic Knudsen number given from the wavelength λ as Kn=x_mfp/λ goes towards one.

In fact, measurements on plane sound wave propagation in rarefied noble gases have indicated that the Burnett model gives a better descrip-tion than the Navier-Stokes-Fourier at such high frequencies [27, 63].

The two models depart significantly in their predictions of sound speed and sound wave absorption at Kn∼0.1, and the Burnett model agrees very well with measurements of sound propagation in noble gases up to Kn∼1.

As it is mathematically very tough to get to even the Burnett level, approximations of even higher order can instead be found through par-ticular assumptions on the form of f . Pioneering work for plane sound waves, where f was assumed to be of the form of an infinitesimal forced plane wave around an equilibrium state, was done by Wang Chang and Uhlenbeck [64] (later collected in [65]). It was later found that even higher-order approximations to the Boltzmann equation paradoxically gives a poorer agreement with experiments than the Burnett model [27].

A very high-order approximation to the Boltzmann equation was later found for the propagation of plane sound waves, of the form of a power series in Kn up toO(Kn³²)[66]. However, the series coefficients increase so rapidly that the series diverges unless Kn is small. This is an example of an asymptotic series, which is non-convergent unless a parameter tends to a certain limit (in this case Kn→0), and which usually is most useful and accurate when truncated to a small number of terms. Still, such series can often be approximated beyond their range of convergence, and in this case the Shanks transformation gave promising results [66].

Similarly, it has been suggested that the Chapman-Enskog expansion f = f⁽⁰⁾+ f⁽¹⁾+ f⁽²⁾+. . . itself may be asymptotic [57, Ch. 15]. If so, truncating the expansion earlier could give a better solution than truncating it later. For sound wave propagation, it seems that the Burnett model gives the best agreement, although it is only significantly better than the Navier-Stokes-Fourier model for about an order of magnitude in Kn. The agreement between the Burnett model and measurements in this single case might also be only a fortunate coincidence and cannot be taken as absolute proof that the Burnett model is generally superior.

Since the Navier-Stokes-Fourier model may be derived independ-ently either from continuum theory, the results of the Chapman-Enskog expansion to first order can be trusted. However, the Burnett model, or any higher-order models for that matter, cannot be found from any other derivation, and have therefore historically been viewed with some suspicion [67]. Since the Burnett model differs very little from the Navier-Stokes-Fourier model at low Kn, the difference is negligible in most practical cases.

Other approaches than the Chapman-Enskog expansion can also be

72 Chapter 3 The kinetic theory of gases

used to find macroscopic equations from the Boltzmann equation. Some of the resulting models have been seen to at least agree well with meas-urements of the speed of sound over the entire range of Kn [68].

3.9 Boltzmann’s H -theorem

One thermodynamic quantity which has not been discussed yet in this chapter is entropy. It was shown by Boltzmann himself that a quantityH can be found from the distribution function f which has many of the same properties as thermodynamic entropy.Hcan only evolve in one direction, and it reaches an extremum when the system is at an equilibrium. It was later shown for ideal gases that His proportional to the entropy.

The first step in findingHis to see from the chain rule that

∂

∂tf ln f = (1+ln f)^{∂ f}_∂t.

This is valid for any derivative in the Boltzmann equation, not just∂/∂t.

Thus, multiplying the Boltzmann equation with(1+ln f), it thus becomes ∂ Integrating this over velocity space, we find

∂

Equation (3.60) is like a conservation equation for the quantity f ln f , but with a source term on the right side. Using the BGK collision operator, the right side can be shown to be

 for f⁽⁰⁾and using the mass and energy conservation properties ofΩ(f). The last inequality follows from the identity(1−x)ln x ≤ 0 for x> 0.

For x=1 (i.e. f = f⁽⁰⁾), it is zero.

3.9 Boltzmann’sH^{-theorem 73}

Thus, (3.60) is equivalent to

∂H

∂t + ^∂H_α

∂x_α ≤0, (3.62)

where

H = f ln f dξ, Hα= ξ_αf ln f dξ. (3.63) Here,Hα is the flux ofH, similarly to howρu is the flux of ρ. From the inequality, we see thatH is not necessarily conserved like mass in the continuity equation, but will decrease if the system is not at equilibrium.

However, as mentioned previously, the inequality in (3.62) becomes an equality at equilibrium. This means thatHwill decrease until the system reaches an equilibrium, where Hreaches its lowest value. This is very similar to how thermodynamic entropy increases until the system reaches an equilibrium state.

In fact, for an ideal gasHis proportional to the entropy densityρs [58, Entropy density,ρs Entropy (“mixedupness”) per physical volume inJ/K m³

69],

ρs= −^kB

mH. (3.64)

However, for a non-ideal gas, where the equation of state is affected by intermolecular forces, this equality does not hold [69].

The inequality (3.62) can also be shown from Boltzmann’s original collision operator. In fact, it is an important criterion for any collision operator, in addition to the conservation criteria discussed in section 3.5, as it states that molecular collisions will invariably drive the distribution of particles towards an equilibrium.

4 The lattice Boltzmann method

In the last chapter we derived the Boltzmann equation and saw that the familiar equations of fluid mechanics follow. In practice, is is extremely hard to find analytical solutions for the Boltzmann equation, except in trivial cases like the spatially homogeneous example in section 3.5 and other simplified cases [70].

In fact, it is also extremely hard to find solutions for the general equa-tions of fluid mechanics, so they are simplified in almost every case. In engineering fluid mechanics, the fluid is often considered incompressible (i.e. the densityρ is considered constant). In acoustics, viscosity is usu-ally neglected and the equations are linearised so that the flow field is considered as a small perturbation around a rest state.

However, if we somehow could find a solution for the Boltzmann equation, we would simultaneously be finding a solution to the less general but more familiar equations that follow from it. Since it is usually too difficult to attack the Boltzmann equation analytically, we must try to solve it numerically instead.

When discretising most transport equations, it is sufficient to discretise in only physical space and time. With the Boltzmann equation, however, the main variable f is a function of coordinates in physical space, velocity space, and time. We must therefore discretise it in two separate steps. First, we restrict the continuous space of velocitiesξ to a finite discrete set ξ_i, a set which should ideally be as small as possible. Then, we simultaneously discretise in space and time. The result of this discretisation process can be quite conveniently implemented on a computer as the lattice Boltzmann method.

There are many approaches to this discretisation, but in this chapter we will emphasise clarity and brevity over generality. Throughout, as necessary, we will refer to articles with other, more general derivations.

In this chapter, we will derive the simplest and most common variety of the LB method: The isothermal, ideal gas, forceless variety. As the

Isothermal fluid A fluid with constant

temperature fluid is isothermal with a constant temperature T0, the ideal gas equation

of state is p=ρRT0. From this follows a constant speed of sound and a simplified equation of state which does not involve the temperature,

c²₀= ∂p

∂ρ

= ^kBT0

m ⇒ p=c²₀ρ. (4.1)

74

4.1 The discrete-velocity Boltzmann equation 75

Comparing this with the physical isentropic equation of state (2.23) and speed of sound (2.24), we find that this isothermal equation of state corresponds to the physical assumption ofγ=1. From (2.47), this itself implies an infinite number of inner degrees of freedom in the molecules that make up the gas.

4.1 The discrete-velocity Boltzmann equation

The first step in discretising the Boltzmann equation is to discretise velo-city space. One very general method for this is based on approximating

f⁽⁰⁾using a truncated basis of Hermite polynomials and a Gauss-Hermite Hermite polynomials An orthogonal polynomial

quadrature [72, 73]. The order of the quadrature determines the number of velocitiesξi required. With sufficiently high orders, this method can preserve the behaviour of the Boltzmann equation to arbitrary level in the Chapman-Enskog expansion [72]. However, higher levels require larger numbers of velocities, which makes the lattice Boltzmann method more difficult to implement and more resource demanding.

Instead of this general method, we will use a mathematically simpler method in this derivation. The first step is to approximate the Maxwell-Boltzmann distribution (3.18) by expanding it up toO(u²),

f⁽⁰⁾(x, ξ, t) = ^ρ

Here, (4.1) has been used, and terms of O(u³) have been neglected.

While stopping atO(u²)may seem somewhat arbitrary, the following subsections will motivate this choice.

Next, we discretise velocity space, restrictingξ to a finite set of velo-citiesξ_i. Thus, the distribution function f(x, ξ, t)becomes f_i(x, t), repres-enting the density at(x, t)of particles with velocityξ_i. We also replace the coefficient e^{−ξαξα/2c20}/(2πc²₀)^3/2in front of the expansion above with a single weighting coefficient w_i, ending up with the classic [74] discrete equilibrium distribution

This is arguably the optimally stable isothermal polynomial discrete equilibrium distribution [75].

We will see in the following subsection how the velocity sets defined

Velocity set

A discrete set of velocity vectorsξiand accompanying weighting coefficientswi

76 Chapter 4 The lattice Boltzmann method

byξ_i and w_imust be constrained in order to reproduce hydrodynamics correctly.

Having discretised velocity space and using the the discrete analogue of the BGK operator (3.22), the Boltzmann equation becomes the discrete-velocity Boltzmann equation (DVBE),

∂ f_i

∂t +ξ_iα∂ f_i

∂x_α = −¹_τ^f_i−f_i⁽⁰⁾



. (4.3)